Hurtigt voksende hierarki

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Det hurtigt voksende hierarki (også kaldet det udvidede Grzegorczyk-hierarki ) er en familie af hurtigt voksende funktioner, der er indekseret efter ordtal . Det mest berømte specielle tilfælde af et hurtigt voksende hierarki er Loeb -Weiner hierarkiet.

Definition

Et hurtigt voksende hierarki er defineret af følgende regler

$f_{0}(n)=n+1$ ( kan generelt være enhver voksende funktion), $f_{0}(n)$
$f_{\alpha +1}(n)=f_{\alpha }^{n}(n)=\underbrace {f_{\alpha }(f_{\alpha }(\cdots (f_{\alpha } (f_{\alpha }(} _{n{\mbox{ gange))}n))\cdots ))$ ,
$f_{\alpha }(n)=f_{\alpha [n]}(n)$ hvis grænsen ordinal, $\alfa$
- hvor er det n'te element i den fundamentale sekvens etableret for en eller anden grænseordinal . $\alpha[n]$ $\alfa$
- Der er forskellige versioner af det hurtigt voksende hierarki, men den mest kendte er Loeb-Weiner hierarkiet, hvor de grundlæggende sekvenser for grænseordtaler skrevet i Cantor normalform er defineret af følgende regler:
$\omega [n]=n$ ,
$(\omega ^{\alpha _{1))+\omega ^{\alpha _{2))+\cdots +\omega ^{\alpha _{k-1))+\omega ^{\ alpha _{k)))[n]=\omega ^{\alpha _{1}}+\omega ^{\alpha _{2}}+\cdots +\omega ^{\alpha _{k-1} }+\omega ^{\alpha _{k}}[n]$
- for , ${\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{k))$
$\omega ^{\alpha +1}[n]=\omega ^{\alpha }n$ ,
$\omega ^{\alpha [n]=\omega ^{\alpha [n]}$ hvis grænsen ordinal, $\alfa$
$\varepsilon _{0}[0]=0$ og . $\varepsilon _{0}[n+1]=\omega ^{\varepsilon _{0}[n]}=\omega \uparrow \uparrow n$

Fundamentale sekvenser for grænseordtaler ovenfor er givet i artiklerne om Veblen- funktioner og Buchholz-funktioner $\varepsilon_{0}$

Eksempler

$f_{1}(n)=f_{0}^{n}(n)=\underbrace {f_{0}(f_{0}(\cdots (f_{0}(f_{0}(} _{n\quad f_{0}}n))\cdots ))=2\gange n$ ,

$f_{2}(n)=f_{1}^{n}(n)=\underbrace {f_{1}(f_{1}(\cdots (f_{1}(f_{1}(} _{n\quad f_{1}}n))\cdots ))=2^{n}\ gange n$ .

For funktioner indekseret med endelige ordinaler , $\alfa >1$

$f_{m}(n)>2\uparrow ^{m-1}n$ .

Især for n =10:

$f_{3}(10)=f_{2}^{10}(10)\ca. \underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}} }}}}}} _{10\quad {\text{ten}}}\approx 10\uparrow \uparrow 11$ ,

$f_{4}(10)=f_{3}^{10}(10)\approx \left.{\begin{matrix}&&\underbrace {10^{10^{10^{\cdots {^ {10^{10^{3.489}}}}}}}} \\&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}}}}}}} } \\&&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489} )))))))} \\&&10\quad {\text{ti}}\end{matrix}}\right\}{\text{10 nedre krøllede seler}}\ca. 10\uparrow \uparrow \uparrow 11$ ,

$f_{m}(10)\approx 10\uparrow ^{m-1}11$ .

Således svarer allerede den første transfinite ordinal til grænsen for Knuths pilnotation . $\omega$

Det berømte Graham-tal er mindre end . $f_{\omega +1}(64)$

På grund af definitionens enkelhed og klarhed bruges det hurtigt voksende hierarki til at analysere forskellige notationer til at skrive store tal .

	Knuth notation	Conway notation	Bowers notation
notationsgrænse	$\omega$	$\omega ^{2}$	$\omega^\omega$
eksempler	$f_{\omega }(10)\ca. 10\uparrow ^{9}11=10\uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 11$	$f_{\omega ^{2}}(n)\ca. \underbrace {n\rightarrow n\rightarrow n\cdots \rightarrow n} _{n+1\quad \rightarrow }$	$f_{\omega ^{\omega }}(n)\ca. \underbrace {\{n,n,n,\cdots ,n,n\}} _{n+2\quad n's}$
eksempler	$f_{\omega +1}(10)\approx \left.{\begin{matrix}&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&\underbrace {\quad \quad \;\;\vdots \quad \quad \;\;} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&{\text{9 pile}}\end{matrix}}\right\}{\text{10}}$	$f_{\omega ^{2}+1}(n)\ca. \left.{\begin{matrix}&&\underbrace {n\rightarrow n\rightarrow \cdots n\rightarrow n} \\&&\underbrace {n\højrepil n\højrepil \cdots n\højrepil n} \\&&\underbrace {\quad \quad \quad \;\;\vdots \quad \quad \quad \;\;} \\&&\underbrace {n \rightarrow n\rightarrow \cdots n\rightarrow n} \\&&{\text{n+1 pile}}\end{matrix}}\right\}{\text{n}}$	$f_{\omega ^{\omega }+1}(n)\ca. \venstre.{\begin{matrix}&&\underbrace {\{n,n,n,\cdots ,n,n\)) \\&&\underbrace {\{n,n,n,\cdots ,n,n\)) \\&&\underbrace {\quad \quad \quad \;\;\vdots \quad \quad \quad \;\ ;} \\&&\underbrace {\{n,n,n,\cdots ,n,n\)) \\&&{\text{n+2 n'er))\end{matrix))\right\}{\ tekst{n}}$

Ovenstående definition definerer et hurtigt voksende hierarki op til . For yderligere vækst kan du bruge Veblen-funktionen og andre endnu mere kraftfulde notationer for ordtaler [1] . $\varepsilon _{0}=\omega \uparrow \uparrow n$

Eksempler

$f_{0}(n)=n+1$
$f_{1}(n)=2n$
$f_{2}(n)=2^{n}n$
$f_{3}(n)>2\uparrow \uparrow n$ (se Knuth pil notation )
$f_{4}(n)>2\uparrow \uparrow \uparrow n$
$f_{m}(n)>2\uparrow ^{m-1}n$
${\displaystyle f_{\omega }(n)>2\uparrow ^{n-1}n=\{n,n,n-1\))$ (se Bowers massiv notation )
$f_{\omega +1}(n)=\{n,n,1,2\}\ca. G(n)$ (se Graham nummer )
${\displaystyle f_{\omega +2}(n)=\{n,n,2,2\))$
${\displaystyle f_{\omega +m}(n)=\{n,n,m,2\))$
${\displaystyle f_{\omega 2}(n)=\{n,n,n,2\))$
${\displaystyle f_{\omega 2+1}(n)=\{n,n,1,3\))$
${\displaystyle f_{\omega 3}(n)=\{n,n,n,3\))$
$f_{\omega {m}}(n)=\{n,n,n,m\}$
$f_{\omega {m+k}}(n)=\{n,n,k,m+1\}$
$f_{\omega ^{2}}(n)=\{n,n,n,n\}$
$f_{\omega ^{3}}(n)=\{n,n,n,n,n\}$
$f_{\omega ^{m}}(n)=\{n,n,n,...,n\}$ (m gange)
$f_{\omega ^{\omega }}(n)=\{n,n,n,...,n\}$ (n gange) $=\{n,n(1)2\}$
$f_{\omega ^{\omega +1}}(n)=\{n,n(1)1,2\}$
$f_{\omega ^{\omega +2}}(n)=\{n,n(1)1,1,2\}$
$f_{\omega ^{\omega 2}}(n)=\{n,n(1)1(1)2\}$
$f_{\omega ^{\omega 2+1}}(n)=\{n,n(1)1(1)1,2\}$
$f_{\omega ^{\omega 3}}(n)=\{n,n(1)1(1)1(1)2\}$
$f_{\omega ^{\omega ^{2}}}(n)=\{n,n(2)2\}$
$f_{\omega ^{\omega ^{3}}}(n)=\{n,n(3)2\}$
${\displaystyle f_{\omega ^{\omega ^{m))}(n)=\{n,n(m)2\))$
${\displaystyle f_{\omega ^{\omega ^{\omega ))}(n)=\{n,n[1,2]2\))$ (se Bird's Array Notation )
${\displaystyle f_{\omega ^{\omega ^{\omega ^{2))))(n)=\{n,n[1,1,2]2\))$
${\displaystyle f_{\omega ^{\omega ^{\omega ^{\omega ))))(n)=\{n,n[1[2]2]2\))$
$f_{\epsilon _{0}}(n)=\{n,n[1{\backslash }2]2\}$
$f_{\omega ^{\omega ^{\epsilon _{0}+1))}(n)=\{n,n[1[1{\backslash }2]2{\backslash }2] 2\}$
${\displaystyle f_{\omega ^{\omega ^{\omega ^{\epsilon _{0}+1))))(n)=\{n,n[1[1[1{\backslash }2] 2{\backslash }2]2{\backslash }2]2\))$
$f_{\epsilon _{1}}(n)=\{n,n[1{\backslash }3]2\}$
$f_{\epsilon _{2}}(n)=\{n,n[1{\backslash }4]2\}$
$f_{\epsilon _{\omega }}(n)=\{n,n[1{\backslash }1,2]2\}$
$f_{\epsilon _{\omega ^{\omega }}}(n)=\{n,n[1{\backslash }1[2]2]2\}$
${\displaystyle f_{\epsilon _{\epsilon _{0))}(n)=\{n,n[1{\backslash }1[1{\backslash }2]2]2\))$
$f_{\epsilon _{\epsilon _{\epsilon _{0))))(n)=\{n,n[1{\backslash }1[1{\backslash }1[1{\backslash' }2]2]2]2\}$
$f_{\zeta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }2]2\}$
$f_{\epsilon _{\zeta _{0}+1}}(n)=\{n,n[1{\backslash }2{\backslash }2]2\}$

$f_{\epsilon _{\zeta _{0}+\omega }}(n)=\{n,n[1{\backslash }1,2{\backslash }2]2\}$

$f_{\epsilon _{\zeta _{0}+\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2]} 2{\backslash }2]2\}$
${\displaystyle f_{\epsilon _{\zeta _{0}2}}(n)=\{n,n[1{\backslash }1{[1{\backslash }1{\backslash }2]}2 {\backslash }2]2\))$
$f_{\epsilon _{\epsilon _{\zeta _{0}+1))}(n)=\{n,n[1{\backslash }1{[1{\backslash }2{\ omvendt skråstreg }2]}2{\backslash }2]2\}$
$f_{\zeta _{1}}(n)=\{n,n[1{\backslash }1{\backslash }3]2\}$
$f_{\zeta _{\omega }}(n)=\{n,n[1{\backslash }1{\backslash }1,2]2\}$
$f_{\zeta _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }2]2]2\ }$
$f_{\zeta _{\zeta _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash}1{\backslash }2 ]2]2\}$
$f_{\eta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2]2\}$
${\displaystyle f_{\varphi (4,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }1{\backslash }2]2\))$
${\displaystyle f_{\varphi (m,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\cdots }1{\backslash }2]2\))$ (m omvendte skråstreg)
${\displaystyle f_{\varphi (\omega ,0)}(n)=\{n,n[1[2{\neg }2]2]2\))$
$f_{\epsilon _{\varphi (\omega ,0)+1))(n)=\{n,n[1{\backslash }2[2{\neg }2]2]2\}$
$f_{\zeta _{\varphi (\omega ,0)+1))(n)=\{n,n[1{\backslash }1{\backslash }2[2{\neg }2] 2]2\}$
$f_{\eta _{\varphi (\omega ,0)+1))(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[2{ \neg }2]2]2\}$
${\displaystyle f_{\varphi (\omega ,1)}(n)=\{n,n[1[2{\neg }2]3]2\))$
${\displaystyle f_{\varphi (\omega ,2)}(n)=\{n,n[1[2{\neg }2]4]2\))$
${\displaystyle f_{\varphi (\omega ,\omega )}(n)=\{n,n[1[2{\neg }2]1,2]2\))$
$f_{\varphi (\omega ,\epsilon _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }2]2]2 \}$
$f_{\varphi (\omega ,\zeta _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }1{\backslash } 2]2]2\}$
$f_{\varphi (\omega ,\varphi (\omega ,0))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2 ]2]2]2\}$
${\displaystyle f_{\varphi (\omega ,\varphi (\omega ,\varphi (\omega ,0)))}(n)=\{n,n[1[2{\neg }2]1[1 [2{\neg }2]1[1[2{\neg }2]2]2]2]2\))$
${\displaystyle f_{\varphi (\omega +1,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }2]2\))$
$f_{\varphi (\omega +2,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }1{\backslash }2]2\}$
${\displaystyle f_{\varphi (\omega 2,0)}(n)=\{n,n[1[2{\neg }2]1[2{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\omega ^{2},0)}(n)=\{n,n[1[3{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\omega ^{3},0)}(n)=\{n,n[1[4{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\omega ^{\omega },0)}(n)=\{n,n[1[1,2{\neg }2]2]2\))$
$f_{\varphi (\omega ^{\omega ^{\omega )),0)}(n)=\{n,n[1[1[2]2{\neg }2]2]2 \}$
$f_{\varphi ({\epsilon _{0)),0)}(n)=\{n,n[1[1[1{\backslash }2]2{\neg }2]2] 2\}$
$f_{\Gamma _{0}}(n)=\{n,n[1[1{\backslash }2{\neg }2]2]2\}$
$f_{\Gamma _{1}}(n)=\{n,n[1[1{\backslash }2{\neg }2]3]2\}$
${\displaystyle f_{\Gamma _{\Gamma _{0))}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1[1{\backslash } 2{\neg }2]2]2]2\))$
${\displaystyle f_{\varphi (1,1,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }2]2\))$
$f_{\varphi (1,2,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }1{\backslash }2 ]2\}$
$f_{\varphi (2,0,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1{\backslash }2{\neg }2]2]2\}$
${\displaystyle f_{\varphi (\omega ,0,0)}(n)=\{n,n[1[2{\backslash }2{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\Gamma _{0},0,0)}(n)=\{n,n[1[1[1[1{\backslash }2{\neg }2]2] 2{\backslash }2{\neg }2]2]2\))$
${\displaystyle f_{\varphi (1,0,0,0)}(n)=\{n,n[1[1{\backslash }3{\neg }2]2]2\))$
${\displaystyle f_{\varphi (1,0,0,0,0)}(n)=\{n,n[1[1{\backslash }4{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\omega )))}(n)=\{n,n[1[1{\backslash }1,2{\neg }2]2] 2\}\ca. TRÆ(n)$ (se TRÆ(3) )
${\displaystyle f_{\psi (\Omega ^{\Omega ^{\psi (\Omega ^{\Omega )))))}(n)=\{n,n[1[1{\backslash }1[ 1[1{\omvendt skråstreg }2{\neg }2]2]2{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\Omega )))}(n)=\{n,n[1[1{\backslash }1{\backslash }2{\neg }2 ]2]2\}$
${\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{2))})}(n)=\{n,n[1[1{\backslash }1{\backslash }1{ \backslash }2{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\Omega ^{\omega ))})}(n)=\{n,n[1[1[2{\neg }2]2{ \neg }2]2]2\}$
${\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\psi (\Omega ^{\Omega ^{\Omega )))))})(n)=\{n,n [ 1[1[1[1[1{\backslash }1{\neg }2{\neg }2]2]2{\neg }2]2{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\Omega ^{\Omega ))})}(n)=\{n,n[1[1[1{\backslash }2{\neg }2]2{\neg }2]2]2\}$
${\displaystyle f_{\psi (\Omega _{2})}(n)=\{n,n[1[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+1)}(n)=\{n,n[1{\backslash }2[1{\neg }3]2]2\))$
$f_{\psi (\Omega _{2}+\omega )}(n)=\{n,n[1{\backslash }1,2[1{\neg }3]2]2\}$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega }))}(n)=\{n,n[1{\backslash }1[1[1{\ omvendt skråstreg }2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\omega ))))}(n)=\{n,n[1{\backslash }1[ 1[1{\omvendt skråstreg }1,2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega ))))}(n)=\{n,n[1{\backslash }1[ 1[1{\backslash }1{\backslash }2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega ^{\Omega ))}))}(n)=\{n,n[1{ \backslash }1[1[1[1{\neg }2{\neg }2]2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega _{2}))}(n)=\{n,n[1{\backslash }1[1[1{\neg }3]2]2[1{\neg }3]2]2\))$
$f_{\psi (\Omega _{2}+\Omega )}(n)=\{n,n[1{\backslash }1{\backslash }2[1{\neg }3]2] 2\}$
$f_{\psi (\Omega _{2}+\Omega ^{2})}(n)=\{n,n[1{\backslash }1{\backslash}1{\backslash }2[ 1{\neg }3]2]2\}$
$f_{\psi (\Omega _{2}+\Omega ^{\omega })}(n)=\{n,n[1[2{\neg }2]2[1{\neg } 3]2]2\}$
$f_{\psi (\Omega _{2}+\Omega ^{\Omega })}(n)=\{n,n[1[1{\backslash }2{\neg }2]2[ 1{\neg }3]2]2\}$
$f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }3] 3]2\}$
${\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2})))}(n)= \{n,n[1[1{\neg }3]1[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+1)))}(n )=\{n,n[1[2{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}( \Omega _{2}))))}(n)=\{n,n[1[1[1{\neg }3]2{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}2)}(n)=\{n,n[1[1{\neg }4]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}3)}(n)=\{n,n[1[1{\neg }5]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}\omega )}(n)=\{n,n[1[1{\neg }1,2]2]2\))$
$f_{\psi (\Omega _{2}\psi (0))}(n)=\{n,n[1[1{\neg }1[1{\backslash }2]2]2 ]2\}$
${\displaystyle f_{\psi (\Omega _{2}\psi (\Omega ^{\Omega }))}(n)=\{n,n[1[1{\neg }1[1[1{ \backslash }2{\neg }2]2]2]2]2\))$
$f_{\psi (\Omega _{2}\psi (\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1[1{\ neg }3]2]2]2]2\}$
${\displaystyle f_{\psi (\Omega _{2}\Omega )}(n)=\{n,n[1[1{\neg }1{\backslash }2]2]2\))$
$f_{\psi (\Omega _{2}\Omega ^{\Omega })}(n)=\{n,n[1[1{\neg }1[1{\backslash }2{\ neg }2]2]2]2\}$
${\displaystyle f_{\psi (\Omega _{2}\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1 {\neg }3]2]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}^{2})}(n)=\{n,n[1[1{\neg }1{\neg }2]2]2\))$
$f_{\psi (\Omega _{2}^{\omega })}(n)=\{n,n[1[1[2{\backslash }_{3}2]2]2] 2\}$
$f_{\psi (\Omega _{2}^{\Omega _{2))}(n)=\{n,n[1[1[1{\backslash }_{2}2{ \ omvendt skråstreg }_{3}2]2]2]2\}$
${\displaystyle f_{\psi (\Omega _{3})}(n)=\{n,n[1[1[1{\backslash }_{3}3]2]2]2\))$
${\displaystyle f_{\psi (\Omega _{3}^{\Omega _{3)))}(n)=\{n,n[1[1[1[1{\backslash }_{3} 2{\backslash }_{4}2]2]2]2]2\))$
$f_{\psi (\Omega _{4})}(n)=\{n,n[1[1[1[1{\backslash }_{4}3]2]2]2]2 \}$
${\displaystyle f_{\psi (\Omega _{\omega })}(n)=\{n,n[1[2{\backslash }_{1,2}2]2]2\))$
$f_{\psi (\Omega _{\psi (\Omega )})}(n)=\{n,n[1[2{\backslash }_{1[1{\backslash }2]2 }2]2]2\}$
$f_{\psi (\Omega _{\psi (\Omega _{\psi (\Omega )})})}(n)=\{n,n[1[2{\backslash }_{1 [2{\backslash _{1[1{\backslash }2]2}}2]2}2]2]2\}$
$f_{\psi (\Omega _{\Omega })}(n)=\{n,n[1[2{\backslash }_{1{\backslash }2}2]2]2\} =(0,0,0)(1,1,1)(2,1,1)(3,1,0)[n]$ (se Bashicu Matrix System )
$f_{\psi ({\Omega _{\Omega _{2))})}(n)=(0,0,0)(1,1,1)(2,1,1)(3 ,1,0)(1,1,0)(2,2,1)(3,2,1)(4,2,0)[n]$
$f_{\psi ({\Omega _{\Omega _{\omega ))})}(n)=(0,0,0)(1,1,1)(2,1,1)( 3,1,0)(1,1,1)[n]$
$f_{\psi ({\Omega _{\Omega _{\Omega ))})}(n)=(0,0,0)(1,1,1)(2,1,1)( 3,1,0)(1,1,1)(2,1,1)(3,1,0)[n]$
$f_{\psi (\psi _{I}(0))}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0 )(2,0,0)[n]$

Se også

Noter

↑ Kerr, Josh Mind blown: det hurtigt voksende hierarki for lægmænd - også kendt som enorme tal . Hentet 7. oktober 2016. Arkiveret fra originalen 13. juli 2019. (ubestemt)

Links

Buchholz, W.; Wainer, S.S. (1987). "Sandsynligvis beregnelige funktioner og det hurtigt voksende hierarki". Logic and Combinatorics , redigeret af S. Simpson, Contemporary Mathematics, Vol. 65, AMS, 179-198.
Cichon, EA & Wainer, SS (1983), The slow-growing and the Grzegorczyk hierarchies , The Journal of Symbolic Logic bind 48 (2): 399–408, ISSN 0022-4812 , DOI 10.2307/2273557
Gallier, Jean H. (1991), Hvad er så specielt ved Kruskals sætning og ordinalen Γ 0 ? En undersøgelse af nogle resultater i bevisteori , Ann. Rent æble. Logic T. 53(3): 199–260, doi : 10.1016/0168-0072 ( >http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA290387, <91)90022-E PDF'er: del 1 2 3 . (Især del 3, afsnit 12, s. 59-64, "Et glimt af hierarkier af hurtige og langsomt voksende funktioner".)
Girard, Jean-Yves (1981), Π 1 2 -logik. I. Dilators , Annals of Mathematical Logic bind 21 (2): 75-219, ISSN 0003-4843 , DOI 10.1016/0003-4843(81)90016-4
Lob, MH; Wainer, S.S. (1970), "Hierarchies of number theoretical functions", Arch. Matematik. Logik , 13. Rettelse, Arch. Matematik. Logik , 14 , 1971 _ _ _ _ _ _ _ _ _
Promel, HJ; Thumser, W.; Voigt, B. "Hurtigt voksende funktioner baseret på Ramsey-sætninger", Discrete Mathematics , v.95 n.1-3, s. 341-358, dec. 1991 doi : 10.1016/0012-365X(91)90346-4 .
Wainer, S.S. (1989), " Langsomt voksende versus hurtigt voksende ". Journal of Symbolic Logic 54 (2): 608-614.

Store tal
Tal	google Shannon nummer Googolplex Skæv nummer Moser nummer Graham nummer TRÆ(3) Rayo nummer
Funktioner	Ackermann funktion Veblen funktion Psi-funktioner af Buchholz tetration Pentation Hyperoperator Hurtigt voksende hierarki Langsomt voksende hierarki Hardy hierarki
Notationer	Steinhaus-Moser notation Knuth pil notation Conway pil notation Massiv Bowers-notation